{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou tput" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "M aple Output" -1 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 3 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle19 " -1 201 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 212 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle6" -1 214 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle7" -1 215 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 20" -1 219 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle26" -1 225 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle27" -1 226 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#################### ###############################" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 40 "# Compute configurations of type 2, n, m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# works if m is small" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# 2,6,2, c3=1.6 -> no symmetry, emb edded" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# 2,4,3, c3=0.7 -> no symmetry, embbeded" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# #################################################" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# choose values of n,m" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 21 "n:=6;m:=2;Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\" \"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 27 "# necksizes - do not touch " }}}{EXCHG {PARA 212 "> \+ " 0 "" {MPLTEXT 1 -1 24 "c3:='c3';c2:=1;c1:='c1';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c3GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1GF$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "# compute c1 as a function of c3" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 63 "solve(2*c1^2+n*(n-1)*c2^2+m*(m-1 )*c3^2-2*n*c1*c2-n*m*c2*c3,c1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,& \"\"$\"\"\"*$,(\"\"'!\"\"*&F(F%%#c3GF%F%*$)F+\"\"#F%F)#F%F.F%,&F$F%F&F )" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 11 "c1:=[%][1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G,&\"\"$\"\"\"*$,(\"\"'!\"\"*&F*F '%#c3GF'F'*$)F-\"\"#F'F+#F'F0F'" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 51 "# compute logarithmic growths to check embeddedness " }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 47 "Q1:=-2*c1;Q2:=2*c1-n *c2;Q3:=n*c2-m*c3;Q4:=m*c3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q1G, &\"\"'!\"\"*&\"\"#\"\"\",(F&F'*&F&F*%#c3GF*F**$)F-F)F*F'#F*F)F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q2G,$*&\"\"#\"\"\",(\"\"'!\"\"*&F*F (%#c3GF(F(*$)F-F'F(F+#F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q3G ,&\"\"'\"\"\"*&\"\"#F'%#c3GF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#Q4G,$*&\"\"#\"\"\"%#c3GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 50 "evalf(solve(e valf(Q1-F'6#$\"54&4T3'G^Dv " 0 "" {MPLTEXT 1 -1 75 "# choose value of c3, s hould be in the above interval for embedded surfaces" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 8 "c3:=1.6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c3G$\"#;!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# necksizes" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 18 "print([c1, c2,c3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"5g'pb=F!R!)>S!#>\"\" \"$\"#;!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# logarithm ic growths" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 21 "evalf([Q1, Q2,Q3,Q4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$!5?$R6Pa!ygR!)!#>$\" 5?$R6Pa!ygR?F&$\"#G!\"\"$\"#KF+" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 80 "# computate the balanced configurations using a poly nomial differential equation" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 78 "P[1]:=z^2-1;P[2]:=z^n+sum(a[k]*z^k,k=0..n-1);P[3]:=z^m+sum(b[ k]*z^k,k=0..m-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"\",& *$)%\"zG\"\"#F'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6# \"\"#,0*$)%\"zG\"\"'\"\"\"F-&%\"aG6#\"\"!F-*&&F/6#F-F-F+F-F-*&&F/F&F-) F+F'F-F-*&&F/6#\"\"$F-)F+F;F-F-*&&F/6#\"\"%F-)F+F@F-F-*&&F/6#\"\"&F-)F +FEF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"$,(*$)%\"zG\" \"#\"\"\"F-&%\"bG6#\"\"!F-*&&F/6#F-F-F+F-F-" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 66 "for i from 1 to 3 do Pp[i]:=diff(P[i],z);Ppp[i] :=diff(Pp[i],z);od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#PpG6#\"\"\" ,$*&\"\"#F'%\"zGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$PppG6#\"\" \"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#PpG6#\"\"#,.*&\"\"'\"\" \")%\"zG\"\"&F+F+&%\"aG6#F+F+*(F'F+&F0F&F+F-F+F+*(\"\"$F+&F06#F5F+)F-F 'F+F+*(\"\"%F+&F06#F:F+)F-F5F+F+*(F.F+&F06#F.F+)F-F:F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%$PppG6#\"\"#,,*&\"#I\"\"\")%\"zG\"\"%F+F+*&F' F+&%\"aGF&F+F+*(\"\"'F+&F16#\"\"$F+F-F+F+*(\"#7F+&F16#F.F+)F-F'F+F+*( \"#?F+&F16#\"\"&F+)F-F6F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#PpG 6#\"\"$,&*&\"\"#\"\"\"%\"zGF+F+&%\"bG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$PppG6#\"\"$\"\"#" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 135 "eq:=collect(expand(c1^2*Ppp[1]*P[2]*P[3]+c2^2*Ppp[2 ]*P[1]*P[3]+c3^2*Ppp[3]*P[1]*P[2]-c1*c2*Pp[1]*Pp[2]*P[3]-c2*c3*Pp[2]*P p[3]*P[1]),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#eqG,8*&,&*&$\"4++ ++++++[%!#=\"\"\"&%\"bG6#F,F,F,*&$\"4KR6Pa!ygR7F+F,&%\"aG6#\"\"&F,F,F, )%\"zG\"\"(F,F,*&,**($\"4KR6Pa!yg>TF+F,F3F,F-F,F,$\"%#f\"!\"#!\"\"*&$ \"5++++++++39F+F,&F.6#\"\"!F,F,*&$\"4kyAu3h:#zWF+F,&F46#\"\"%F,F,F,)F8 \"\"'F,F,*&,,*&$\"$/#FBF,F-F,FB*($\"4kyAu3h:#fdF+F,FLF,F-F,F,*($\"5KR6 Pa!yg>@\"F+F,F3F,FFF,F,*&$\"4'zT8J;M#)=(*F+F,&F46#\"\"$F,F,*&$\"$7*FAF ,F3F,FBF,)F8F6F,F,*&,.*($\"$?\"FBF,F3F,F-F,FB*($\"5kyAu3h:#f@\"F+F,FLF ,FFF,F,*&$\"#IFHF,FFF,FB*($\"4'zT8J;M#))R*F+F,FinF,F-F,F,*&$\"5GdX[EF+F,&F4F/F,F,*($\"5'zT8J;M#))>9F+F,FinF,FFF,F ,*&$\"$_\"FAF,FinF,FB*($\"5GdX[FAF,F`qF,FB*($FerFBF,FapF,F-F,F,*($FPFHF,FinF ,FFF,FB*($\"5g'pb=F!R!yU#F+F,F`qF,FFF,F,F,F8F,F,*&$\"$7&FAF,F\\sF,FB*( $\"#;FBF,F`qF,F-F,F,*(FfsF,F\\sF,FFF,F,*($FcpFHF,FapF,FFF,FB" }}} {EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 157 "eqalist:=\{\}:for i from 1 to n do eqalist:=\{op(eqalist),coeff(eq,z^(2+n+m-2-i))\};od:alist:= \{\}:for i from 1 to n do alist:=\{op(alist),a[i-1]\};od:eqalist;alist ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(,&*&$\"4++++++++[%!#=\"\"\"&%\" bG6#F)F)F)*&$\"4KR6Pa!ygR7F(F)&%\"aG6#\"\"&F)F),**($\"4KR6Pa!yg>TF(F)F 0F)F*F)F)$\"%#f\"!\"#!\"\"*&$\"5++++++++39F(F)&F+6#\"\"!F)F)*&$\"4kyAu 3h:#zWF(F)&F16#\"\"%F)F),.*($\"#?FAF)F0F)F?F)F;*&$\"5g'pb=F!R!)>EF(F)& F1F,F)F)*($\"5'zT8J;M#))>9F(F)&F16#\"\"$F)F?F)F)*&$\"$_\"F:F)FSF)F;*($ \"5GdX[@ \"F(F)F0F)F?F)F)*&$\"4'zT8J;M#)=(*F(F)FSF)F)*&$\"$7*F:F)F0F)F;,.*($\"$ ?\"F;F)F0F)F*F)F;*($\"5kyAu3h:#f@\"F(F)FEF)F?F)F)*&$\"#IFAF)F?F)F;*($ \"4'zT8J;M#))R*F(F)FSF)F*F)F)*&$\"5GdX[ " 0 "" {MPLTEXT 1 -1 27 "sola:=solve(eqalist,alist);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solaG<(/&%\"aG6#\"\"$,(*&$\"5%*eNr8B]])R$!#>\"\"\"&% \"bG6#F0F0!\"\"*($\"58`h2/T\\`pjF/F0F1F0&F26#\"\"!F0F0*&$\"5*)eC>'))>( op>F/F0)F1F*F0F4/&F(6#\"\"#,.*&$\"5o?e_uhH3!e\"F/F0F8F0F4$\"5GP3h-JC(R 0*!#?F0*&$\"5G&zpZ/sp#QD#F/F0)F8FBF0F0*($\"5sxG`PDyS8fF/F0F8F0FRF0F4/&F(6#FN ,($\"5s\"pK%=@P>aNF/F0*&$\"5i!QA.A^2M9$F/F0F8F0F4*&$\"5olFx$zW(*QK$F/F 0FRF0F0/&F(F3,.*($\"5k%y\"HvC[#fu%F/F0F1F0FVF0F4*($\"53)z#>ti%)\\CJFIF 0F1F0F8F0F4*&$\"5@oSmu!)[x#G%!#@F0F1F0F0*&$\"5mgvg-i&p&p\\FIF0F>F0F0*( $\"5mVK(pvv)*>Y%F/F0F8F0F>F0F0*&$\"5VX\"R!4+.TmiFIF0)F1\"\"&F0F4/&F(6# Fgp,$*&$\"5`4a!HyHYSh$F/F0F1F0F4/&F(F9,6$\"5RIK+W&)zDTFIF0FVF0 F4*($\"5#yc/)yl&y3E\"F/F0F8F0FRF0F0*($\"5r*ff@Ui\"pbdF/F0FVF0FRF0F0*&$ \"5BDg&z)QUuJMFIF0FMF0F4*($\"5G`8%f^$ppMKF/F0F8F0FMF0F4*&$\"5$>QC@8%G* z4\"F/F0)F8F*F0F4*&$\"5jC%)=R9*3fz$FIF0)F1\"\"'F0F0" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 175 "eqblist:=\{\}:for i from 1 to m do eq blist:=\{op(eqblist),numer(normal(subs(sola,coeff(eq,z,i-1))))\}:od:eq blist;blist:=\{\}:for i from 1 to m do blist:=\{op(blist),b[i-1]\};od: blist;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$,>*($\"5&)QG$4.I#G^:!#=\" \"\")&%\"bG6#\"\"!\"\"#F))&F,6#F)F/F)F)*($\"5)G#*[5;$zPX5!#na>%H&)Gb6F(F)F+F)F0F)F9*&$\"5z5::O;aj?u!#?F)F0F)F )*&$\"5Cuk*Rz*)Hqm%!#>F)F+F)F9*&$\"5Qh(ycq[a%[NF(F))F+F8F)F9*&$\"5U`' \\%\\.kAq^FDF))F+\"\"$F)F9*&$\"5oO2J=758YHFDF))F1\"\"'F)F9*&$\"5j%p]I- =%=_DFDF)F7F)F)*($\"53rz2*=4/,'=F6F)FLF)F0F)F)$\"5gRlHX&)3C:*)!#@F9*&$ \"5;X%z)z(4WOd#F(F)F*F)F)*($\"5OqlKg9[pU5F(F)F+F)F7F)F)*($\"5K(3@TqV[n A\"F(F)F+F)FQF)F),$*&F1F),6*&$\"5CTA2Y)pSq-%F(F)F*F)F)*($\"5A$*QnOvk(o *[F(F)F+F)F0F)F9*&$\"5CaB^/^SXx&)FDF)F7F)F)*&$\"5_K\"3s.'y_rTF(F)F+F)F 9*&$\"5K(3@TqV[nA\"F(F)FQF)F9$\"5DlZ@>?5)pc\"FDF9*&$\"5bS6,tQq/)*\\FDF )F0F)F)*($\"5We%4Y;*)*QVHF6F)F*F)F0F)F9*($\"5)*QTkV&39v>\"F6F)F+F)F7F) F)*&$\"5!403g)HH12:F6F)FLF)F)F)F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< $&%\"bG6#\"\"!&F%6#\"\"\"" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 41 "solb:=evalf(solve(evalf(eqblist),blist));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solbG6><$/&%\"bG6#\"\"\"$\"\"!F-/&F)6#F-$!5+++++++++ 5!#><$F'/F/$!5:;>?Y1wLUA!#?<$F'/F/$\"57^SVs L)4\")\\'F=<$/F(^#$!5u'*>'3qVa'R6F3/F/$!5,81$zqVD*)4$F=<$/F(^#$\"5u'*> '3qVa'R6F3FG<$/F/^$$!5k7TTQS6*eT\"F=$!5p5R(oA]\"=$G'F=/F(^$$!5P()eehf) 3Te)F=$\"5r5R(oA]\"=$G'F=<$FP/F(^$$\"5P()eehf)3Te)F=$!5r5R(oA]\"=$G'F= <$/F/^$FR$\"5p5R(oA]\"=$G'F=/F(^$FXF[o<$F^o/F(^$FinFZ<$/F/^$$!5AaQhd+m 'p@\"F=$!5(pG+&fCpST>F=/F(^$$!5zXhQU*RLIy)F=$\"5(pG+&fCpST>F=<$/F(^$$ \"5zXhQU*RLIy)F=F\\pFho<$/F/^$FjoFbp/F(^$F`pF\\p<$Fjp/F(^$FgpFbp<$/F(^ $$!5'))4qpF**Qx'HF=$\"5(4ZQDiR)3S]F=/F/^$$!50%*3]JI0\"=C'F8$!5s]zyx(* \\nCUF8<$/F(^$$\"5'))4qpF**Qx'HF=$!5(4ZQDiR)3S]F=Fhq<$/F/^$Fjq$\"5s]zy x(*\\nCUF8/F(^$FdqFcr<$Ffr/F(^$FarFfq<$/F(^#$!5%QX<%\\%GNdy&F=/F/$\"5) Q;.q$pD:$\\\"F=<$Fds/F(^#$\"5%QX<%\\%GNdy&F=<$/F/$\"5_8$3arnQX?#F=/F($ !5ryNq<09YX7F3<$/F($\"5ryNq<09YX7F3F]t<$/F($!5ghRc]]XRR7F3/F/$\"5'ehRc ]]XRR#F=<$/F($\"5ghRc]]XRR7F3F[u<$/F(^#$!5#oR]Ms'QNuIF=/F/$\"5k()*eEyR 5JY$F=<$/F(^#$\"5#oR]Ms'QNuIF=Fgu<$/F($!5BbZl!\\Op)[]F3/F/$\"5*\\va1\\ Op)[SF3<$Fcv/F($\"5BbZl!\\Op)[]F3" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 75 "# select true solutions, namely such that P1, P2 and P3 have no common root" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 219 "SOLB:=\{\}:for i from 1 to nops([solb]) do sol:=subs([solb][i],so la) union [solb][i]:if abs(subs(sol,resultant(P[1],P[2],z)*resultant(P [2],P[3],z)*resultant(P[1],P[3],z)))>10^(-6) then SOLB:=\{op(SOLB),sol b[i]\}:fi:od:SOLB;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<1<$/&%\"bG6#\" \"\"^#$!5u'*>'3qVa'R6!#>/&F'6#\"\"!$!5,81$zqVD*)4$!#?<$/F&$F1F1/F/$!5: ;>?Y1wLUAF4<$F6/F/$\"57^SVsL)4\")\\'F4<$/F& ^#$\"5u'*>'3qVa'R6F-F.<$/F&^$$!5'))4qpF**Qx'HF4$\"5(4ZQDiR)3S]F4/F/^$$ !50%*3]JI0\"=C'F;$!5s]zyx(*\\nCUF;<$/F&^#$!5%QX<%\\%GNdy&F4/F/$\"5)Q;. q$pD:$\\\"F4<$Fen/F&^#$\"5%QX<%\\%GNdy&F4<$/F/$\"5_8$3arnQX?#F4/F&$!5r yNq<09YX7F-<$/F&$\"5ryNq<09YX7F-F^o<$/F&^$$\"5'))4qpF**Qx'HF4$!5(4ZQDi R)3S]F4FP<$/F/^$FR$\"5s]zyx(*\\nCUF;/F&^$FLF]p<$F`p/F&^$F[pFN<$/F&^#$! 5#oR]Ms'QNuIF4/F/$\"5k()*eEyR5JY$F4<$/F&^#$\"5#oR]Ms'QNuIF4F^q" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 15 "solb:=op(SOLB):" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 13 "nops([solb]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "# select solution. Real of pure imaginary solutions give symme tric configurations." }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 10 " choice:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'choiceG\"\"'" }}} {EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 -1 13 "solb[choice];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"bG6#\"\"\"^$$!5'))4qpF**Qx'H!#?$\"5(4 ZQDiR)3S]F,/&F&6#\"\"!^$$!50%*3]JI0\"=C'!#@$!5s]zyx(*\\nCUF6" }}} {EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 52 "sol:=subs([solb][choice], sola) union [solb][choice];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG <*/&%\"bG6#\"\"\"^$$!5'))4qpF**Qx'H!#?$\"5(4ZQDiR)3S]F./&F(6#\"\"!^$$! 50%*3]JI0\"=C'!#@$!5s]zyx(*\\nCUF8/&%\"aG6#\"\"$^$$\"5kT!zvX:U@o)F.$!5 vi&*>K\"HBN%=!#>/&F=F)^$$\"57.;v)>j>R\\&F8$\"5%y-(Ge1]XKJF8/&F=6#\"\"& ^$$\"5(zjAu>haD2\"FE$!58j_4\"4G6:#=FE/&F=6#\"\"#^$$\"5yLUli$RV$f#*F.$! 4d*Q&[ho:!e;F./&F=6#\"\"%^$$\"51l2PfQzz)>$FE$!5&Hf+q%GEc:')F./&F=F3^$$ \"5s,$Gnv%fHg6F8$\"5V;G!HU(\\ad7F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# check equations has been solved" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 13 "subs(sol,eq);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2*&^$$!\"\"!#>$\"\"!F*\"\"\")%\"zG\"\"(F+F+*&^$F)$F+F( F+)F-\"\"'F+F+*&^$$\"\"#F($F'!#=F+)F-\"\"&F+F+*&^$F)$!\"&F(F+)F-\"\"%F +F+*&F%F+)F-\"\"$F+F+*&^$F&$!\"$!#?F+)F-F7F+F+*&^$$\"#=!#@$!\"#FIF+F-F +F+^$F)$F+FOF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# compute the points of the configuration" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 54 "for i from 1 to 3 do p[i]:=[solve(subs(sol,P[i]))];o d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"\"7$F'!\"\"" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"#7(^$$!5R()y%f_\"pnP`!#?$ \"5@K!H$e?p?2E!#>^$$!5x:le#4oqyI$F,$!50B(e!Rq*y#F,^$$!58.(>5E%=)o\"pF =$\"5=M6m20\\#p1\"F,^$$\"5g8r=c\"R'yqmF=$!5'3>y&)*G!*3^;F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"$7$^$$\"5-\"zX^!H(e/@$!#?$!5,sA !=)H.TiMF,^$$!5e@pv\"GO(>FC!#@$!5'*)>O2k1ywd\"F," }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 -1 27 "p1:=p[1];p2:=p[2];p3:=p[3];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#p1G7$\"\"\"!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p2G7(^$$!5R()y%f_\"pnP`!#?$\"5@K!H$e?p?2E!#>^$$!5x:l e#4oqyI$F)$!50B(e!Rq*y#F)^$$!58.(>5E%=)o\"pF:$\"5=M6m20\\#p1\"F)^$$\" 5g8r=c\"R'yqmF:$!5'3>y&)*G!*3^;F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#p3G7$^$$\"5-\"zX^!H(e/@$!#?$!5,sA!=)H.TiMF)^$$!5e@pv\"GO(>FC!#@$!5'* )>O2k1ywd\"F)" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 25 "# plot \+ the configuration " }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 -1 12 "wi th(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoo rds has been redefined\n" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 77 "plot1:=complexplot(p1,axes=none,style=point,color=green,scaling=co nstrained):" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 75 "plot2:=com plexplot(p2,axes=none,style=point,color=red,scaling=constrained):" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 76 "plot3:=complexplot(p3,axes =none,style=point,color=blue,scaling=constrained):" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 27 "display(plot1,plot2,plot3);" }}{PARA 13 "" 1 "" {GLPLOT2D 536 536 536 {PLOTDATA 2 "6)-%'CURVESG6%7$7$$\"\" \"\"\"!$F*F*7$$!\"\"F*F+-%'COLOURG6&%$RGBGF+$\"*++++\"!\")F+-%&STYLEG6 #%&POINTG-F$6%7(7$$!5R()y%f_\"pnP`!#?$\"5@K!H$e?p?2E!#>7$$!5x:le#4oqyI $F@$!50B(e!Rq*y#F@7$$!58.(>5E%=)o\"pFQ$\"5=M6m20\\#p1\"F@7$$\"5g8r=c \"R'yqmFQ$!5'3>y&)*G!*3^;F@-F06&F2F3F+F+F6-F$6%7$7$$\"5-\"zX^!H(e/@$F@ $!5,sA!=)H.TiMF@7$$!5e@pv\"GO(>FCFQ$!5'*)>O2k1ywd\"F@-F06&F2F+F+F3F6-% *AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"Fd p-%%FONTG6#%(DEFAULTG-%%VIEWG6$FipFip" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# check that configuration i s balanced and non-degenerate" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 10 "nn:=2+n+m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG\"#5" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 19 "QQ:=matrix(nn,4,0):" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 to 2 do QQ[i ,1]:=-c1:QQ[i,2]:=c1:od:for i from 1 to n do QQ[2+i,2]:=-c2;QQ[2+i,3]: =c2;od:for i from 1 to m do QQ[2+n+i,3]:=-c3;QQ[2+n+i,4]:=c3;od:" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 30 "pp:=vector(nn):FF:=vector( nn):" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 144 "for i from 1 to \+ nn do FF[i]:=sum(sum(QQ[i,kk]*QQ[jj,kk]/(pp[i]-pp[jj]),jj=1..i-1)+sum( QQ[i,kk]*QQ[jj,kk]/(pp[i]-pp[jj]),jj=i+1..nn),kk=1..4);od:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 157 "SUBS:=\{pp[1]=p1[1],pp[2]=p1[2] \}: for i from 1 to n do SUBS:=\{op(SUBS),pp[2+i]=p2[i]\};od:for i fro m 1 to m do SUBS:=\{op(SUBS),pp[2+n+i]=p3[i]\};od:print(SUBS);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<,/&%#ppG6#\"\"\"F(/&F&6#\"\"#!\"\"/&F &6#\"\"$^$$!5R()y%f_\"pnP`!#?$\"5@K!H$e?p?2E!#>/&F&6#\"\"%^$$!5x:le#4o qyI$F5$!50B(e!Rq*y#F5/&F&6#\"\"(^$$!58.(>5E%=)o \"pFR$\"5=M6m20\\#p1\"F5/&F&6#\"\")^$$\"5g8r=c\"R'yqmFR$!5'3>y&)*G!*3^ ;F5/&F&6#\"\"*^$$\"5-\"zX^!H(e/@$F5$!5,sA!=)H.TiMF5/&F&6#\"#5^$$!5e@pv \"GO(>FCFR$!5'*)>O2k1ywd\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# check that forces are zero" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 49 "for i from 1 to nn do evalf(subs(SUBS,FF[i]));od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"#!#>$!\"%!#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"#5!#>$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"\"!F%$!#9!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"#!#?$\"\"# !#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"(!#?$\"\"\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"&!#>$\"\"\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!#?F%$!\"\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $!\"\"!#=$\"\"'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"\"!F%$\"\" \"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"\"\"!#=$!\"&!#>" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# compute Jacobian matrix A " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 37 "A:=evalf(subs(SUBS,ja cobian(FF,pp))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "# confi guration is nondegenerate if ony two singular values are zero ( say le ss than 10^-6 )" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 16 "singul arvals(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,^$$\"5([uQ)pZBa[R!#<$! 5xl.ie5-\"H%=!#Q^$$\"5wbILj(H.!p5F'$!5PVK9U[NXHIF*^$$\"5#\\qU**3&yDPF! #=$!5VB\"30-Hu&oa!#P^$$\"5blll;ZlD+\\.)HH!#O^$$\"5-h%G6z mkt7\"F3$!5w>IMko3W@CF6^$$\"5rMdNGMx&yQ'!#>$!5.:cb`['yy\"eF*^$$\"5Yc=u 6gQFEZFE$!5,\"RL&zGZBiUF<^$$\"5\"R2Aj " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# export the configuration as a list of poi nts" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 25 "lprint([0.,op(p),o p(q)]);" }}{PARA 6 "" 1 "" {TEXT -1 433 "[0., table([(1)=[1, -1],(2)=[ -.53376769152594788739+2.6072069205832903221*I, -.33078706809258651577 -1.2185154173905872305*I, -.13580722120774469700+.77020638591240665897 *I, -.69731676637464020587e-1-.27897019580324006250*I, -.6916881842610 1970313e-1+.10669249050766113418*I, .66707863915618711360e-1-.16510890 289857819086*I],(3)=[.32104587290514579102-.34624103298180227201*I, -. 24271973628175692158e-1-.15776780664073619896*I]]), q]" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 19 "lprint([c1,c2,c3]);" }}{PARA 6 " " 1 "" {TEXT -1 31 "[4.0198039027185569660, 1, 1.6]" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "# export the configuration in latex picture format (the result needs \+ to be edited somewhat...)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "xmin:=-1:ymin:=0:xmax:=1:ymax:=0:for i from 1 to n do xmin:=min(x min,Re(p2[i])):xmax:=max(xmax,Re(p2[i])):ymin:=min(ymin,Im(p2[i])):yma x:=max(ymax,Im(p2[i])):od:for i from 1 to m do xmin:=min(xmin,Re(p3[i] )):xmax:=max(xmax,Re(p3[i])):ymin:=min(ymin,Im(p3[i])):ymax:=max(ymax, Im(p3[i])):od:height:=max(xmax-xmin,ymax-ymin):" }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 0 366 "lprint(begin(picture)(100*(xmax-xmin),100*h eight)(100*xmin,100*(ymin+ymax-height)/2));lprint(setlength(unitlength )(0.5/(xmax-xmin)));lprint(put(-100,0));lprint(\{bullet\});lprint(put( 100,0));lprint(\{bullet\});for i from 1 to n do lprint(put(100*Re(p2[i ]),100*Im(p2[i])));lprint(\{circ\});od;for i from 1 to m do lprint(put (100*Re(p3[i]),100*Im(p3[i])));lprint(\{star\});od;" }}{PARA 6 "" 1 " " {TEXT -1 70 "begin(picture)(200,382.57223379738775526)(-100,-121.851 54173905872306)" }}{PARA 6 "" 1 "" {TEXT -1 44 "setlength(unitlength)( .25000000000000000000)" }}{PARA 6 "" 1 "" {TEXT -1 11 "put(-100,0)" }} {PARA 6 "" 1 "" {TEXT -1 8 "\{bullet\}" }}{PARA 6 "" 1 "" {TEXT -1 10 "put(100,0)" }}{PARA 6 "" 1 "" {TEXT -1 8 "\{bullet\}" }}{PARA 6 "" 1 "" {TEXT -1 49 "put(-53.376769152594788739,260.72069205832903221)" }} {PARA 6 "" 1 "" {TEXT -1 6 "\{circ\}" }}{PARA 6 "" 1 "" {TEXT -1 50 "p ut(-33.078706809258651577,-121.85154173905872305)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{circ\}" }}{PARA 6 "" 1 "" {TEXT -1 49 "put(-13.580722120 774469700,77.020638591240665897)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{circ \}" }}{PARA 6 "" 1 "" {TEXT -1 50 "put(-6.9731676637464020587,-27.8970 19580324006250)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{circ\}" }}{PARA 6 "" 1 "" {TEXT -1 49 "put(-6.9168818426101970313,10.669249050766113418)" } }{PARA 6 "" 1 "" {TEXT -1 6 "\{circ\}" }}{PARA 6 "" 1 "" {TEXT -1 49 " put(6.6707863915618711360,-16.510890289857819086)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{circ\}" }}{PARA 6 "" 1 "" {TEXT -1 49 "put(32.1045872905 14579102,-34.624103298180227201)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{star \}" }}{PARA 6 "" 1 "" {TEXT -1 50 "put(-2.4271973628175692158,-15.7767 80664073619896)" }}{PARA 6 "" 1 "" {TEXT -1 6 "\{star\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 225 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT -1 0 "" }}}{MARK "57 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }